As long as your games are all from legitimate sources -- either retail boxed versions or from reputable download sites like Amazon, Big Fish, GOG, Gamersgate, or Adventure Shop -- it's unlikely that one of your games would cause you not to be able to get on the Internet. It's more likely that your ISP had temporary problems or there was some other glitch with your computer.

However if you downloaded a game from a suspicious website, it may have been infected with a virus or some other malware that interfered with your Internet connection.